2. Since H is the midpoint of arc AG >> HI.HM=HN.HE= HA ^2And quad(NIME) is cyclic.

3. Note that MN is radical line of circle(ABC) and circle(NIME)In circle(BICE) , F is the intersection of chords BC and IE >> FI.FE=FB.FC Since power of F to circle (NIME)=power of F to circle(ABC)So F must lie on radical axis MN of circles (ABC) and circle (NIME)